Example of retract which is not closed

Question

I know that retract of Hausdorff space must be a closed subset.

So I was thinking any non Hausdorff space will work as example. So I thought Cofinite Topology space with infinitely many elements.

So finite element set are closed .If I choose infinite set but How to show there exist retract.I cannot visualise this.

Please give me some hint

Any help will be appreciated

Answer 1

No need to visualise anything: As a concrete instance, take $X=\Bbb Z$ in the cofinite topology and take $A=\Bbb N = \{n \in \Bbb Z: n \ge 0\}$, which is not closed as its complement is not finite.

Take $f(n)=|n|$ which maps $X$ to $A$ and is the identity on $A$. $f$ is continuous because the inverse image of a finite set (the only non-trivial closed subsets) is finite (so closed) because $f$ is at most $2$-to-$1$. So $f$ is a retraction of a non-Hausdorff $X$ onto a non-closed subset.

I chose these sets for the easy description, but if $X$ is infinite cofinite and $A$ is a subset of $X$ with $|A| = |X\setminus A| = |X|$, then we can also find a $2$-to-$1$ (so continuous) retraction of $X$ onto $A$, giving lots of similar examples.

Answer 2

Any constant function is continuous, so just find a space where a point is not closed then retract to that point.

Answer 3

A more geometric example is given by the real line with doubled origin. It retracts to the line (mapping both copies of the origin to one of them) but this is not a closed subspace.